Reversible Boolean networks I: distribution of cycle lengths

We consider a class of models describing the dynamics of N Boolean variable s, where the time evolution of each depends on the values of K of the other variables. Previous work has considered models with dissipative dynamics. Here, we consider time-reversible models, which necessarily have the proper ty that every possible point in the state space is an element of one and on ly one cycle. The orbits can be classified by their behavior under time rev ersal. The orbits that transform into themselves under time reversal have p roperties quite different from those that do not; in particular, a signific ant fraction of latter-type orbits have lengths enormously longer than orbi ts: that are time-reversal symmetric. For large K and moderate N, the vast majority of points in the state space are on one of the time-reversal singl et orbits, and a random hopping model gives an accurate description of orbi t lengths. However, for any finite K, the random hopping approximation fail s qualitatively when N is large enough(N >> 2(2K)). As in the dissipative c ase, when K is large, typical orbit lengths grow exponentially with N, wher eas for small enough K, typical orbit lengths grow much more slowly with N. The numerical data are consistent with the existence of a phase transition at which the average orbit length grows as a power of N at a value of K be tween 1.4 and 1.7. However, in the reversible models, the interplay between the discrete symmetry and quenched randomness can lead to enormous fluctua tions of orbit lengths and other interesting features that are: unique to t he reversible case. (C) 2001 Elsevier Science B.V. All rights reserved.